I am attempting to arrive at the Black-Scholes formula for my own understanding. I can accept one can use the risk-free distribution & rate, so I am attempting to use the distrution to arrive at the result rather than PDEs. Here is what I have:
$$\lim\limits_{b \to \infty}\int_a^b pdf(s)(s-K)ds$$

Since the initial price is $S_0$ I popped that in with $s$.

$$\lim\limits_{b \to \infty}\int_a^b pdf(s*S_0)(s*S_0-K)ds$$

Where $a = K / S_0$.

And since B-S formula choose the log normal I will use that for the pdf:

I should probably mention here that I have been lazily mentally substituting $\sigma\sqrt{t}$ anytime I see $\sigma$.

Since $b$ is going to $\infty$,$\Phi(\beta)$ is going to $1$. At this point we should also try getting rid of $\mu$. Since $S_0e^{rt}$ is the expected forward price we can set that equal to $e^{\mu+\sigma^2/2}$ and get:

$$\mu = ln(S_0) + rt - \sigma^2/2$$

As for the second integral, we are just going to get a difference of two log-normal CDFs (I think) times $K$. The first will again be $1$ since $b$ is going to $\infty$, so we get $K(1-LNCDF(a))$.

My result is thus $S_0e^{rt}(1-\Phi(a)) - K(1-LNCDF(a))$ This doesn't look too familiar so I am wondering what I am doing wrong. Thanks!